Why do I hear a rattling noise coming from the engine when I start the car? on 2003 Dodge Neon

Every time I start the car, the engine makes a rattling noise that really can be heard from inside the car. When I change it into reverse, the engine stops rattling.
When I open the hood and the car is in either Park or Drive, the rattling gets worse when I press down on the motor. If I pull up on the motor, the rattling noise goes almost away. When I drive and I am at at stop light or stop sign, the engine continues to rattle. Once the engine gets hot from driving around, the rattling noise coming from the engine goes away.
I checked the motor mounts, well at least I think I did and they looked ok.
I took it to the dealer and asked them how much a diagnostic would cost. I told that maybe the motor mounts may have to be changed. their estimate was at around 600 dollars...
Any ideas of what it could be?

Asked by for the 2003 Dodge Neon
i had a recent problem like that, all the same noises, i also checked the motor mounts and all that and everything was fine, the problem was just that my exhaust cover was rattling against my exhaust, and a quick bolt fixed it.
1 more answer
The dealer is probably right about the engine mounts but I would get another estimate, I see a five star rated repair shop on Discount Car Care